Let μ be a signed Radon measure in the Kato class and define a Schrödinger type operator ${\cal H}^{\lambda \mu}=\frac{1}{2}(-\Delta)^{\frac{\alpha}{2}}+\lambda \mu \ \text{on}\ ℝ^{d}$. We show that its spectral bound $C(\lambda)=-\text{inf}\sigma ({\cal H}^{\lambda \mu})$ is differentiable if α < d ≤ 2α and μ is Green-tight.

Differentiability of Spectral Functions for Symmetric α-Stable Processes

10.1090/s0002-9947-07-04149-9